Abstract A bootstrap percolation process on a graph with n vertices is an ‘infection’ process evolving in rounds. Let $r ge 2$ be fixed. Initially, there is a subset of infected vertices. In each subsequent round, every uninfected vertex that has at least r infected neighbors becomes infected as well and remains so forever. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank one. Assuming that initially every vertex is infected independently with probability $p in (0,1]$ , we provide a law of large numbers for the size of the set of vertices that are infected by the end of the process. Moreover, we investigate the case $p = p(n) = o(1)$ , and we focus on the important case of inhomogeneous random graphs exhibiting a power-law degree distribution with exponent $beta in (2,3)$ . The first two authors have shown in this setting the existence of a critical $p_c =o(1)$ such that, with high probability, if $p =o(p_c)$ , then the process does not evolve at all, whereas if $p = omega(p_c)$ , then the final set of infected vertices has size $Omega(n)$ . In this work we determine the asymptotic fraction of vertices that will eventually be infected and show that it also satisfies a law of large numbers.
具有n个顶点的图上的自举渗透过程是一个以轮为单位演化的“感染”过程。让$r ge 2$固定下来。最初，有一个受感染顶点的子集。在随后的每一轮中，每个至少有r个被感染邻居的未感染顶点也会被感染，并永远保持这种状态。我们考虑下面的图是一个核为秩1的非齐次随机图的情况。假设最初每个顶点都以$p in (0,1]$的概率独立感染，我们提供了一个大数定律，用于表示在过程结束时被感染的顶点集的大小。此外，我们研究了$p = p(n) = o(1)$的情况，并重点研究了指数为$beta in (2,3)$的幂律度分布的非齐次随机图的重要情况。在这种情况下，前两位作者已经证明了一个临界$p_c =o(1)$的存在，这样，在高概率下，如果$p =o(p_c)$，则该过程根本不会进化，而如果$p = omega(p_c)$，则最终受感染顶点集的大小为$Omega(n)$。在这项工作中，我们确定了最终将被感染的顶点的渐近分数，并表明它也满足大数定律。
{"title":"Bootstrap percolation in inhomogeneous random graphs","authors":"Hamed Amini, Nikolaos Fountoulakis, Konstantinos Panagiotou","doi":"10.1017/apr.2023.21","DOIUrl":"https://doi.org/10.1017/apr.2023.21","url":null,"abstract":"Abstract A bootstrap percolation process on a graph with n vertices is an ‘infection’ process evolving in rounds. Let $r ge 2$ be fixed. Initially, there is a subset of infected vertices. In each subsequent round, every uninfected vertex that has at least r infected neighbors becomes infected as well and remains so forever. We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank one. Assuming that initially every vertex is infected independently with probability $p in (0,1]$ , we provide a law of large numbers for the size of the set of vertices that are infected by the end of the process. Moreover, we investigate the case $p = p(n) = o(1)$ , and we focus on the important case of inhomogeneous random graphs exhibiting a power-law degree distribution with exponent $beta in (2,3)$ . The first two authors have shown in this setting the existence of a critical $p_c =o(1)$ such that, with high probability, if $p =o(p_c)$ , then the process does not evolve at all, whereas if $p = omega(p_c)$ , then the final set of infected vertices has size $Omega(n)$ . In this work we determine the asymptotic fraction of vertices that will eventually be infected and show that it also satisfies a law of large numbers.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We propose a modification to the random destruction of graphs: given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon ( J. Austral. Math. Soc. 11 , 1970) and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions for binary caterpillar trees and complete binary trees.
{"title":"A modification of the random cutting model","authors":"Fabian Burghart","doi":"10.1017/apr.2023.22","DOIUrl":"https://doi.org/10.1017/apr.2023.22","url":null,"abstract":"Abstract We propose a modification to the random destruction of graphs: given a finite network with a distinguished set of sources and targets, remove (cut) vertices at random, discarding components that do not contain a source node. We investigate the number of cuts required until all targets are removed, and the size of the remaining graph. This model interpolates between the random cutting model going back to Meir and Moon ( J. Austral. Math. Soc. 11 , 1970) and site percolation. We prove several general results, including that the size of the remaining graph is a tight family of random variables for compatible sequences of expander-type graphs, and determine limiting distributions for binary caterpillar trees and complete binary trees.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider infinitely wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a heavy-tailed distribution in the domain of attraction of a symmetric $alpha$ -stable distribution, where $alphain(0,2]$ may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric $alpha$ -stable distribution having the same $alpha$ parameter as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we extend a recent result of Favaro, Fortini, and Peluchetti (2020) to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric $alpha$ -stable distributions, $alphain(0,2]$ .
{"title":"-Stable convergence of heavy-/light-tailed infinitely wide neural networks","authors":"Paul Jung, Hoileong Lee, Jiho Lee, Hongseok Yang","doi":"10.1017/apr.2023.3","DOIUrl":"https://doi.org/10.1017/apr.2023.3","url":null,"abstract":"\u0000 We consider infinitely wide multi-layer perceptrons (MLPs) which are limits of standard deep feed-forward neural networks. We assume that, for each layer, the weights of an MLP are initialized with independent and identically distributed (i.i.d.) samples from either a light-tailed (finite-variance) or a heavy-tailed distribution in the domain of attraction of a symmetric \u0000 \u0000 \u0000 \u0000$alpha$\u0000\u0000 \u0000 -stable distribution, where \u0000 \u0000 \u0000 \u0000$alphain(0,2]$\u0000\u0000 \u0000 may depend on the layer. For the bias terms of the layer, we assume i.i.d. initializations with a symmetric \u0000 \u0000 \u0000 \u0000$alpha$\u0000\u0000 \u0000 -stable distribution having the same \u0000 \u0000 \u0000 \u0000$alpha$\u0000\u0000 \u0000 parameter as that layer. Non-stable heavy-tailed weight distributions are important since they have been empirically seen to emerge in trained deep neural nets such as the ResNet and VGG series, and proven to naturally arise via stochastic gradient descent. The introduction of heavy-tailed weights broadens the class of priors in Bayesian neural networks. In this work we extend a recent result of Favaro, Fortini, and Peluchetti (2020) to show that the vector of pre-activation values at all nodes of a given hidden layer converges in the limit, under a suitable scaling, to a vector of i.i.d. random variables with symmetric \u0000 \u0000 \u0000 \u0000$alpha$\u0000\u0000 \u0000 -stable distributions, \u0000 \u0000 \u0000 \u0000$alphain(0,2]$\u0000\u0000 \u0000 .","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47692607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a novel Monte Carlo algorithm for the vector consisting of the supremum, the time at which the supremum is attained, and the position at a given (constant) time of an exponentially tempered Lévy process. The algorithm, based on the increments of the process without tempering, converges geometrically fast (as a function of the computational cost) for discontinuous and locally Lipschitz functions of the vector. We prove that the corresponding multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $varepsilon^{-2}$ if the mean squared error is at most $varepsilon^2$ ) and provide its central limit theorem (CLT). Using the CLT we construct confidence intervals for barrier option prices and various risk measures based on drawdown under the tempered stable (CGMY) model calibrated/estimated on real-world data. We provide non-asymptotic and asymptotic comparisons of our algorithm with existing approximations, leading to rule-of-thumb principles guiding users to the best method for a given set of parameters. We illustrate the performance of the algorithm with numerical examples.
{"title":"A Monte Carlo algorithm for the extrema of tempered stable processes","authors":"J. G. González Cázares, Aleksandar Mijatovi'c","doi":"10.1017/apr.2023.1","DOIUrl":"https://doi.org/10.1017/apr.2023.1","url":null,"abstract":"\u0000 We develop a novel Monte Carlo algorithm for the vector consisting of the supremum, the time at which the supremum is attained, and the position at a given (constant) time of an exponentially tempered Lévy process. The algorithm, based on the increments of the process without tempering, converges geometrically fast (as a function of the computational cost) for discontinuous and locally Lipschitz functions of the vector. We prove that the corresponding multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order \u0000 \u0000 \u0000 \u0000$varepsilon^{-2}$\u0000\u0000 \u0000 if the mean squared error is at most \u0000 \u0000 \u0000 \u0000$varepsilon^2$\u0000\u0000 \u0000 ) and provide its central limit theorem (CLT). Using the CLT we construct confidence intervals for barrier option prices and various risk measures based on drawdown under the tempered stable (CGMY) model calibrated/estimated on real-world data. We provide non-asymptotic and asymptotic comparisons of our algorithm with existing approximations, leading to rule-of-thumb principles guiding users to the best method for a given set of parameters. We illustrate the performance of the algorithm with numerical examples.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41909126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Developed sequential order statistics (DSOS) are very useful in modeling the lifetimes of systems with dependent components, where the failure of one component affects the performance of remaining surviving components. We study some stochastic comparison results for DSOS in both one-sample and two-sample scenarios. Furthermore, we study various ageing properties of DSOS. We state many useful results for generalized order statistics as well as ordinary order statistics with dependent random variables. At the end, some numerical examples are given to illustrate the proposed results.
{"title":"Ordering and ageing properties of developed sequential order statistics governed by the Archimedean copula","authors":"Tanmay Sahoo, Nil Kamal Hazra","doi":"10.1017/apr.2023.25","DOIUrl":"https://doi.org/10.1017/apr.2023.25","url":null,"abstract":"\u0000 Developed sequential order statistics (DSOS) are very useful in modeling the lifetimes of systems with dependent components, where the failure of one component affects the performance of remaining surviving components. We study some stochastic comparison results for DSOS in both one-sample and two-sample scenarios. Furthermore, we study various ageing properties of DSOS. We state many useful results for generalized order statistics as well as ordinary order statistics with dependent random variables. At the end, some numerical examples are given to illustrate the proposed results.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44711617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This note corrects an error in the formula to obtain the Whittle index using the Sherman–Morrison formula in Akbarzadeh and Mahajan (2022). Also, some other minor typos are highlighted.
{"title":"Conditions for indexability of restless bandits and an algorithm to compute whittle index – CORRIGENDUM","authors":"N. Akbarzadeh, Aditya Mahajan","doi":"10.1017/apr.2022.77","DOIUrl":"https://doi.org/10.1017/apr.2022.77","url":null,"abstract":"\u0000 This note corrects an error in the formula to obtain the Whittle index using the Sherman–Morrison formula in Akbarzadeh and Mahajan (2022). Also, some other minor typos are highlighted.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41882972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time-homogeneity of the problem considered. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a nonlinear barrier. In the related article of Grandits et al. (Scand. Actuarial J.2, 2007), it has been observed that standard numerical methods break down in certain parameter cases, and no closed-form solution has been found. For these reasons, we offer a new method allowing one to estimate the distance from an arbitrary smooth-enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies—payout on the maximal rate, and constant barrier strategies—by measuring the distance from their performance functions to the value function.
{"title":"Measuring the suboptimality of dividend controls in a Brownian risk model","authors":"J. Eisenberg, Paul Krühner","doi":"10.1017/apr.2023.6","DOIUrl":"https://doi.org/10.1017/apr.2023.6","url":null,"abstract":"\u0000 We consider an insurance company modelling its surplus process by a Brownian motion with drift. Our target is to maximise the expected exponential utility of discounted dividend payments, given that the dividend rates are bounded by some constant. The utility function destroys the linearity and the time-homogeneity of the problem considered. The value function depends not only on the surplus, but also on time. Numerical considerations suggest that the optimal strategy, if it exists, is of a barrier type with a nonlinear barrier. In the related article of Grandits et al. (Scand. Actuarial J.2, 2007), it has been observed that standard numerical methods break down in certain parameter cases, and no closed-form solution has been found. For these reasons, we offer a new method allowing one to estimate the distance from an arbitrary smooth-enough function to the value function. Applying this method, we investigate the goodness of the most obvious suboptimal strategies—payout on the maximal rate, and constant barrier strategies—by measuring the distance from their performance functions to the value function.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45027262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored $delta$ -shock model, $delta ge 1$ , for which the system fails whenever no shock occurs within a $delta$ -length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob.32, 2000), we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system’s lifetime follows a Markov geometric distribution of order $delta$ (a geometric distribution of order $delta$ under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system’s lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored $delta$ -shock model, for which the system fails when no shock occurs within a $delta$ -length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold $gamma >0$ . Similarly, for this mixed model, we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.
假设一个系统受到一系列随机冲击的影响，这些冲击在一定时期内发生。本文研究了离散截尾$delta$ -冲击模型$delta ge 1$，该模型假定连续冲击之间的到达时间由一阶马尔可夫链描述(以及在二项冲击过程下，即连续冲击之间的到达时间具有几何分布)，当从最后一次冲击到$delta$ -长度的时间段内没有发生冲击时，系统失效。利用Chadjiconstantinidis等人引入的马尔可夫链嵌入技术。prop .32, 2000)，我们研究了系统寿命、冲击次数和无冲击发生的周期数的联合分布和边际分布，直至系统失效。得到了这些随机变量的联合概率和边际概率生成函数，并给出了它们的概率质量函数和矩的几个递推式和精确公式。证明了系统的寿命服从阶为$delta$的马尔可夫几何分布(二项设置下阶为$delta$的几何分布)，并服从矩阵几何分布。通过证明系统寿命随机变量的位移服从复合几何分布，给出了系统在二项冲击过程下的一些可靠性特性。最后，我们引入了一种新的混合离散截尾$delta$ -冲击模型，当最后一次冲击在$delta$ -长度的时间内没有发生冲击，或者冲击的幅度大于给定的临界阈值$gamma >0$时，系统就会失效。同样地，对于该混合模型，我们研究了在二项冲击过程下，系统的寿命、冲击次数和不发生冲击的周期数的联合分布和边际分布，直至系统失效。
{"title":"Distributions of random variables involved in discrete censored δ-shock models","authors":"S. Chadjiconstantinidis, S. Eryilmaz","doi":"10.1017/apr.2022.72","DOIUrl":"https://doi.org/10.1017/apr.2022.72","url":null,"abstract":"\u0000 Suppose that a system is affected by a sequence of random shocks that occur over certain time periods. In this paper we study the discrete censored \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -shock model, \u0000 \u0000 \u0000 \u0000$delta ge 1$\u0000\u0000 \u0000 , for which the system fails whenever no shock occurs within a \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -length time period from the last shock, by supposing that the interarrival times between consecutive shocks are described by a first-order Markov chain (as well as under the binomial shock process, i.e., when the interarrival times between successive shocks have a geometric distribution). Using the Markov chain embedding technique introduced by Chadjiconstantinidis et al. (Adv. Appl. Prob.32, 2000), we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system. The joint and marginal probability generating functions of these random variables are obtained, and several recursions and exact formulae are given for the evaluation of their probability mass functions and moments. It is shown that the system’s lifetime follows a Markov geometric distribution of order \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 (a geometric distribution of order \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 under the binomial setup) and also that it follows a matrix-geometric distribution. Some reliability properties are also given under the binomial shock process, by showing that a shift of the system’s lifetime random variable follows a compound geometric distribution. Finally, we introduce a new mixed discrete censored \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -shock model, for which the system fails when no shock occurs within a \u0000 \u0000 \u0000 \u0000$delta$\u0000\u0000 \u0000 -length time period from the last shock, or the magnitude of the shock is larger than a given critical threshold \u0000 \u0000 \u0000 \u0000$gamma >0$\u0000\u0000 \u0000 . Similarly, for this mixed model, we study the joint and marginal distributions of the system’s lifetime, the number of shocks, and the number of periods in which no shocks occur, up to the failure of the system, under the binomial shock process.","PeriodicalId":53160,"journal":{"name":"Advances in Applied Probability","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46101365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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